1. Field of the Invention
The present invention relates to a multivariable statistical method for analyzing images that have been formed of a complex environment such as subsoil for identifying spatial relations between elements of the structure of the environment.
2. Description of the Prior Art
Spatial contiguity analysis (SCA) has been the subject of many publications, notably by:
L. Lebart, 1969, Analyse statistique de la contiguxc3xaftxc3xa9, Pub. Ins. Stat., Paris VIII, 81-112.
Under the designation xe2x80x9cspatial proximity analysisxe2x80x9d, spatial contiguity analysis has been applied notably for seismic data filtering as described by:
Royer, J. J., 1984, Proximity Analysis: a Method for Geodata Processing, in Sciences de la Terre, nxc2x0 20, Proc. of the Int. Coll.: Computers in Earth Sciences for Natural Resources Characterization, April 9-13, Nancy, France, or by
Faraj, A., 1994, Application of Spatial Contiguity Analysis to Seismic Data Filtering. In SECxe2x80x9464th Annual International SEG Meeting, Los Angeles, October. 23018 1994, Expanded abstracts, vol. 1, Paper SP5.7, 15841587.
Spatial contiguity analysis computes a family of linear combination components of the initial variables which minimize the contiguity coefficient known as Geary""s coefficient defining the ratio of the spatial variability to the variance.
The components (referred to as spatial components) correspond to the eigenvectors of the matrix Cxe2x88x921xcex93(h), where C is the variance-covariance matrix of the initial data and xcex93(h) that of the variograms-crossed variograms at the spatial distance h. The spatial components are usually arranged in descending order of the eigenvalues of this matrix.
Generally first components (associated with the low eigenvalues), referred to as regional components, represent the large-scale spatial structures. The last spatial components (associated with the high eigenvalues), referred to as local components, relate to the small-scale spatial structures.
The information xe2x80x9cbornexe2x80x9d by these various components is measured by the associated eigenvalues.
The methodology followed for spatial contiguity analysis is directly modelled on the well-known principal-components analysis (PCA). It has however been observed that the eigenvalues of matrix Cxe2x88x921xcex93(h) are poor measurements of the local variance for analyzing and arranging the spatial components. In fact, during certain analyses, the components associated with the highest eigenvalues are totally meaningless (random noise for example), whereas those corresponding to intermediate eigenvalues seem to better account (at least visually) for both the statistical and the spatial information of the initial variables.
In reality, although calculated from matrices C and xcex93(h) which include both the statistical and spatial interdependences of the data and although measuring the contiguity relations of the factorial components, the eigenvalues represent each an isolated piece of information (ratio of the local variance to the global variance) specific to the spatial component. The eigenvalue is a criterion that is good only for measuring the spatial variability of the spatial component. Unlike the principal-components analysis (PCA) for example, where the eigenvalues (of C) represent the part of the total variance of the data explained by the component, the sum of the eigenvalues obtained with the SCA, which is tr[Cxe2x88x921xcex93(h)], is meaningless. On the other hand, it is preferable to obtain tr[xcex93(h)] because it represents the sum of the local variances of the initial variables. It is therefore necessary to define new criteria in order to quantify the statistical and spatial information borne by the spatial components of the SCA.
It is a multivariable statistical method for analysing data associated (directly or indirectly) with image elements showing physical properties of a complex environment, these images being obtained by exploration of the environment (by means of seismic waves for example) so as to highlight the spatial relations between these elements. The method comprises:
spatial analysis of the data in order to show the spatial properties of events in at least one direction;
application to the data of a spatial contiguity analysis technique in order to best split the data into component spatial structures, and
filtering of the component spatial structures obtained by splitting in order to extract the most pertinent spatial structures.
The method comprises:
identifying the component spatial structures of the data,
separating the spatial structures including eliminating possible redundancies,
forming, from the initial images, of synthetic images or spatial components showing the spatial structures of the data,
marking a typology of the initial images according to the spatial structures shown by the spatial components, and
filtering of the initial images in order to suppress noise and to select at least one identified spatial structures.
The spatial structure of the data is for example determined by a single variable or a pair variables analysis of different variograms of the image elements.
Selection of the component spatial structures is achieved for example by determining the respective contributions thereof to the spatial variability of the data and to the variance of the data, and this selection can be achieved graphically.
The method allows identification of the spatial structures as well as quantification of the filtered information in terms of variance and of spatial variability.
In a general way, the method can be applied to the analysis of any evenly or unevenly distributed spatial data.